My driver side windshield wiper was frozen to the windshield. I forgot to shut them off when I went to work. When I came out and was brushing off the snow I realized that they were not off. Then the driver side windshield wiper did not move when the passenger side did. Now I have to manually push down the driver side wiper. When I turn them on they both go up and only the passenger side goes down

I just solved this problem last night with my '05 Impala, and ultimately the solution was what newton5083 stated above.  Yet, it may also be this (I found it online somewhere):
You have bent or broken the "park tab" on the windshield wiper motor bracket. The correct way to fix this is to remove the wiper arms, wiper linkage and wiper motor from the wiper motor bracket, and install a new bracket.
I just bent the tab back into place a couple times for our daughters 2005 Impala but it kept getting bent down again when they didn't take the time to make sure the wipers were not frozen to the windshield, or tried running the wipers to remove heavy snow on the windshield. So yesterday I fabricated a small piece of metal and bolted that to the bracket in the same place the tab stop was.
If someone wants to access this bracket and tab-
1. Use a ½ inch or 13mm socket & ratchet or wrench to remove nuts on wiper arms.
2. They may make a special tool to remove the arms, I used a screwdriver and hammer to tap bottom of the arm to wedge them apart from the spindle.
3. Two small button-looking clips hold the drivers side cowl cover in place under the windshield. These are two-piece clips, pry up the center, than the outer piece.
4. Unclip the washer hose from cowl cover, than pull out plastic cover.
5. There is a smaller plastic cover with on plastic button head holding that cover in place. Gently pry under that clip to remove that cover.
6. Now you can see the wiper trans and bracket. You should see a small tab on that bracket that looks pushed down and bent or possible a square small hole with some torn off metal where the tab broke off.
7. Remove small plastic cover over the wiper transmission and you can see a small spring-loaded arm underneath the wiper transmission assembly.
8. You need to find something to pry that tab back up so when the wipers are running that spring loaded little arm just snaps past the tab running clockwise. It's when you shut off the wipers that the wiper transmission assembly reverses rotation counter clockwise and the small spring loaded arm underneath hits that tab stop, than the wipers will go all the way down and shut off as they should.
Once that tab stop gets bent out of the way the little spring loaded arm has nothing to stop the wiper transmission assembly so it keeps traveling counter clockwise until some other stop shuts the wipers off straight up on the windshield.

There is a revised bracket part # which is supposed to be more durable, GM part #12368639. Here is a copy of the TSB which points out this problem:
Bulletin No.: 05-08-43-003A
Date: March 10, 2005
TECHNICAL
Subject:
Windshield Wipers Park in Up Position (Replace Windshield Wiper Motor Bracket)
Models:
2005 Chevrolet Impala, Monte Carlo
Supercede:
This bulletin is being revised to make a correction to the Parts Information. Please discard Corporate Bulletin Number 05-08-43-003 (Section 08-Body and Accessories).

Condition
Some customers may comment that the windshield wipers park in the out wipe (vertical) position.
Cause
This condition may be caused by a bent park tab on the windshield wiper motor bracket.

Correction
To correct this condition, replace the wiper motor bracket. It will be necessary to transfer the following parts listed below to the new bracket.
^ Wiper transmission
^ Wiper motor crank arm
^ Wiper motor
1. Remove the windshield wiper motor drive system module.
2. Remove the windshield wiper transmission from the windshield wiper motor bracket.
3. Remove the wiper motor crank arm from the windshield wiper motor.
4. Remove the windshield wiper motor from the bracket.
5. Install the windshield wiper motor to the new bracket.
6. Install the wiper motor crank arm to the windshield wiper motor.
7. Install the windshield wiper transmission to the windshield wiper motor bracket.
8. Install the windshield wiper motor drive system module.

A TSB is not a recall, if vehicle is still under warranty the dealer should be repaired for free. If not, you'll have to pay for the repairs. 

The wipers on our 2005 Impala would also park in the vertical position. I was perplexed b/c the tab on the wiper motor mounting bracket was not bent down out of alignment. As it turned out, I was simply able to lubricate the spring loaded arm mechanism (in this case on top of the wiper motor/transmission), and this corrected the problem.

My 1998 Buick Century encountered the similar situation. The wipers were turned on with snow on the windshield. Both posts were imediately stripped. I believe they are designed to strip out the spline on the arm to prevent damage to the motor or linkage. If it were an older car and you did this, it would have simply stalled the wiper motor with no damage; go figure! I had to buy two new wiper arms at $45 each! Thanks for the new way GM!

Try tightening the nut where the wiper arm meets the pivot, there may be a cover over it.